5490
J. Phys. Chem. B 1998, 102, 5490-5494
Structure and Adsorption of a Hard Sphere Fluid in a Cylindrical and Spherical Pore Filled by a Disordered Matrix: A Monte Carlo Study Y. Duda,*,† S. Sokolowski,‡ P. Bryk,‡ and O. Pizio† Instituto de Quimica de la UNAM, Coyoaca´ n 04510, Me´ xico D.F., Mexico, and Department of Modelling of Physico-Chemical Processes, Marie Curie-Sklodowska UniVersity, Lublin 200-31, Poland ReceiVed: February 11, 1998; In Final Form: May 6, 1998
We have studied a model of a hard sphere fluid adsorbed in a cylindrical and spherical pore filled with a quenched disordered matrix of hard sphere particles using Grand canonical Monte Carlo simulations. The interactions of both matrix and fluid species with pore walls are assumed to be of a hard sphere type. We discuss the adsorption isotherms and density profiles of fluid particles in pores with different microporosities for several values of pore radius. We have observed that, similar to homogeneous microporous media, the adsorption increases with increasing porosity. However, trends of behavior of the isotherms also reflect layering of adsorbed fluid. The data obtained serve as a benchmark for the development of the theory of confined quenched-annealed systems and for computer simulation investigation of models permitting phase transitions in pores.
Introduction The structural and thermodynamic properties of fluids confined in random porous media are of interest for material science, physical chemistry, and some biological problems. There has been much progress in theoretical studies of these quenched-annealed systems during recent years. Computer simulations and extension of liquid-state integral equations to quenched-annealed fluids have been developed. Theoretical work in quenched-annealed continuous systems has been initiated by Madden and Glandt;1,2 more recently, Given and Stell, after detailed analysis of the cluster expansion for the partition function of quenched-annealed systems,1,2 have derived exact Replica Ornstein-Zernike (ROZ) integral equations.3,4 A set of approximations for these equations also has been proposed.4,5 The majority of previous works, see, for example, refs 5-10, have been restricted to the description of homogeneous quenched-annealed fluids. Very recently, studies of inhomogeneous quenched-annealed fluid systems have been initiated. However, only models with hard core type interactions for all species involved have been considered, by using the ROZ equations and computer simulations.11-14 The exploration of phase transitions in confined quenched-annealed systems has not yet been attempted. Moreover, few previous works in this area have been restricted to a slitlike confinement. In this work our principal objective is to investigate the adsorption of a simple fluid in a cylindrical and spherical pore filled with a disordered microporous matrix. Matrix-free pores of cylindrical and spherical shape have been the subject of much research.15-22 In particular, phase transitions have been considered. Methodological tools such as integral equations, density functional approach, and computer simulations have complemented each other in the microscopic description of phenomena in cylindrical and spherical pores. In the present study we restrict ourselves * Address correspondence to this author at the Institute for Condensed Matter Physics, National Academy of Sciences of the Ukraine, Lviv 11, Ukraine. † Instituto de Quimica de la UNAM. ‡ Marie Curie-Sklodowska University.
to the model without attractive interaction such that phase transitions other than crystallization cannot take place. Even at this level of modeling the adsorption isotherms for fluids in cylindrical and spherical pores filled with disordered matrixes are unavailable. We have investigated the system by Grand canonical Monte Carlo (GCMC) simulations. The reason is that the application of the inhomogeneous ROZ equations, used for matrix-filled slitlike pores, is much more difficult for other geometries. On the other hand, the density functional method, successful for matrix-free cylindrical and spherical pores, has not yet been used for any confined quenched-annealed fluids. The results of the presented simulations, beside being of interest on their own, are necessary to begin the development of relevant theoretical approaches. Moreover, these results may serve as a helpful benchmark for further computer simulation studies of phase transitions in matrix-filled cylindrical and spherical pores. This problem seems to represent a challenge for the theory, as well as for simulations, because of physically rich phenomena expected. An interplay of critical behavior due to the interactions of curved pore walls with the criticality of fluids confined in a homogeneous microporous random media may yield unexpectedly interesting characteristics of some materials. Model for a Fluid in a Matrix-Filled Cylindrical Pore In the present work we are considering a model for a hard sphere fluid (f) with the interaction
Uff(r) )
{
∞, r < σf 0, r > σf
(1)
where σf is the diameter of fluid species. The matrix particles of the diameter σm interact in a similar manner
Umm(r) )
{
∞, r < σm 0, r > σm
(2)
The cross-interaction, Ufm(r), is given in the same form in eqs 1 and 2 but with characteristic length σfm, σfm ) 0.5(σf +
S1089-5647(98)01127-4 CCC: $15.00 © 1998 American Chemical Society Published on Web 06/24/1998
Structure and Adsorption of a Hard Sphere Fluid
J. Phys. Chem. B, Vol. 102, No. 28, 1998 5491
TABLE 1: Parameters Used in the Simulation Runs for a Fluid in Cylindrical and Spherical Poresa βµm ) 2
βµm )-1.5
Rc
lc
βµf
〈Nf〉/〈Nm〉
Nm.conf
P
〈Nf〉/〈Nm〉
Nm.conf
P
1.5 1.5 1.5 2.5 2.5 2.5 5.0 5.0 5.0 5.0 5.0 5.0 5.0
30 30 30 25 25 25 15 15 15 15
0 2 4 0 2 4 -1.5 0 2 4 0 2 4
11.5/115.4 31.9/115.4 59.9/115.4 21.7/239.6 65.8/239.6 118.8/239.6 17.1/533.7 48.1/533.7 141/533.7 263/533.7 19.8/219.8 66.1/219.8 124.8/219.8
65 70 70 65 50 50 45 45 48 48 100 100 130
0.73348
45.6/24.6 94.8/24.6 41.9 96/55.5 192.4/55.5 284/55.5 90.8/130.8 218.7/130.8 433.5/130.8 620.6/130.8 86.5/43.4 188.8/43.4 542/43.4
40 55 48 40 40 35 35 35 35 35 85 70 75
0.94163
0.75502 0.7681
0.7157
0.94241 0.94296
0.943
a Notations: the radius of cylinder (or sphere), Rc; length of cylinder, lc; average number of fluid, 〈Nf〉, and matrix, 〈Nm〉, particles; number of matrix configurations, Nm.conf, matrix porosity, P.
σm). Both matrix and fluid species are immersed in a circular cylinder or a sphere with radius Rc. The pore walls are to be assumed impermeable for matrix and for fluid species, that is
Ui(r) )
{
∞, R > Rc 0, R < Rc
(3)
where R is the distance along normal of a particle belonging to species i (i ) f,m) to the pore walls. In this study we restrict ourselves to the simple case of equal diameters of fluid and matrix particles, σm ) σf ) 1. The imaginary experiment we are thinking about is performed as follows. The matrix species in the bulk is characterized by the chemical potential βµm. The matrix particles are adsorbed in a cylinder and attain equilibrium in the density distribution characterized by the density profile Fm(R). This matrix structure becomes quenched at certain conditions. Adsorption of a fluid with the chemical potential βµf is then attempted. After equilibration, the fluid species attain the density profile Ff(R). The density profile describes the structure of adsorbed fluid. Moreover, using Ff(R) and Fm(R), we obtain the normalized adsorption isotherms, Γ*(βµf,βµm)
Γ*(βµf,βµm) )
∫0R dRRFf(R)/[0.5R2c - π6 ∫0R dRRFm(R)] c
c
(4) and the excess normalized adsorption isotherms Γ*ex(βµf,βµm)
Γ/ex(βµf,βµm) )
to enter the matrix-filled pore by using the conventional GCMC method again. In the case of a cylindrical pore the MC simulation cell was a cylinder. The periodic boundary conditions with the minimum image convention were imposed along the axis of the cylinder, that is, i.e., in z-direction. The height of the cylinder was larger for the smaller value of its radius to increase the number of particles; technical details of the simulations are given in the Table 1. In the case of a spherical pore the simulations have been performed in just a single sphere. The matrix and then the fluid inside that cavity are assumed to be in equilibrium with the bulk fluid. In practice, it can be achieved by means of a hole in the cavity wall. However, the mechanism for achieving this equilibrium is not specified in the simulations. Each simulational step consists of attempts to move a particle, as in the common Metropolis MC algorithm, to delete a randomly chosen particle, or to create a particle at a random position. The acceptance of moving has not been lower than 20-30% in all runs. Averaging over at least 35 (at βµm ) -1.5) or 40 (at βµm ) 2) matrix configurations was sufficient to provide density profiles for the fluid species. The length of each of run ranged from 1.5 × 109 to 5 × 109 steps.
∫0R dRR(Ff(R) c
[
Fbf )/ 0.5R2c -
π 6
∫0R dRRFm(R)] c
(5)
where Fbf is the density of bulk fluid at βµf, which can be calculated from the Carnahan-Starling equation of state. Normalization of the adsorption isotherms has been performed to facilitate a comparison of adsorption of a fluid in pores of different diameter and with different microporosities. The microporosity is characterized by the difference between the volume of a cylindrical pore and the volume occupied by matrix particles in that pore. It is given by the denominator in eq 4. In the case of a spherical pore the normalization has been performed similarly. The simulation in Grand canonical ensemble has been performed as follows. First, the matrix has been simulated in the GCMC procedure. After equilibration, several matrix configurations have been selected at random and frozen. In the second step, we have allowed the fluid species
Results and Discussion Let us proceed with the description of the results obtained. We consider first the model in which a cylindrical pore is large, Rc ) 5, intermediate, Rc ) 2.5, or small, Rc ) 1.5, in width. The adsorption isotherms, Γ*(βµf,βµm), for a fluid in cylindrical pores for the case of a dilute matrix characterized by a low value of the chemical potential, βµm ) -1.5, and in the case of a dense matrix at βµm ) 2, are given in Figure 1. The values of matrix porosity per unit volume of the pore space are listed in the Table 1. We observe that the adsorption of a fluid increases with increasing chemical potential βµf. The porosity value remains almost constant in the experiments performed at βµm ) -1.5, P ≈ 0.942, as well as at βµm ) 2, P ≈ 0.755, such that a higher adsorption in a narrower cylindrical pore can be attributed both to a stronger fluid adsorption on the pore walls of a narrower cylinder and to a higher fluid density in the central region of the cylinder, if compared to a wider cylinder. The excess adsorption, Γex*, is higher in a narrower pore than in a wider one (see Table 2). However, we observe that for a given pore width, the adsorption is higher in the pore with higher porosity, that is, in the matrix characterized by a lower value of the chemical potential.
5492 J. Phys. Chem. B, Vol. 102, No. 28, 1998
Duda et al.
Figure 1. Adsorption isotherms, Γ*(βµf,βµm), for a fluid in cylindrical pores for the case of βµm ) -1.5 (upper group of curves) and βµm ) 2 (lower group of curves). Radius of cylinder: Rc ) 5 (circles), 2.5 (crosses), 1.5 (squares).
TABLE 2: Excess Adsorption, Γ/ex, of a Fluid in a Cylindrical Porea
a
βµf
Rc ) 5
Rc ) 2.5
Rc ) 1.5
0 2 4
-0.0716 -0.0572 -0.0310
βµm ) -1.5 -0.0579 -0.0226 0.0289
-0.0363 0.0357 0.1237
0 2 4
-0.2742 -0.3929 -0.4388
βµm ) 2 -0.2765 -0.3826 -0.4110
-0.2761 -0.3660 -0.3610
Notations and the Length of Cylinder Are the Same as In Table 1.
The density profiles of fluid particles in cylindrical pores of different width at βµm ) -1.5 are shown in Figure 2. The contact values for the profiles on the pore walls have been obtained by extrapolation. The contact values and the density close to the pore walls are higher in the narrow pore, Rc ) 1.5, if compared to the wide pore, Rc ) 5. It follows from a comparison of the positions and values of the maxima of the density profiles in pores with different Rc values that the fluid is more structured in the narrower pores. It appears that for low matrix density the fluid-fluid correlations are of major importance in the behavior of the density profiles. The oscillations of the profiles develop with increasing chemical potential of fluid species; they develop more rapidly in narrower pores. The density profiles of fluid particles in cylindrical pores of different width but at βµm ) 2 are shown in Figure 3. Most importantly, for a high-density matrix with the density varied
according to the chosen value of the chemical potential, βµm, the average fluid density is much lower in comparison with the case of a dilute matrix (cf. Figures 2 and 3). For all pores under study, that is, with Rc ) 5, 2.5, or 1.5, we observe that the contact value for the fluid density profile is lower than that for the matrix species. The oscillations of fluid density are seen to develop with increasing chemical potential βµf; however, they are of smaller magnitude compared to the case of adsorption in a dilute matrix. Moreover, we observe that the oscillations of the matrix density profile and of adsorbed fluid density profile are slightly shifted in phase. This shift can be seen for the three pores in question and can be explained by a stronger effect of pore walls compared with the effect of filling of the pore with matrix particles. Stronger effects of the matrix distribution on the fluid distribution in the pore may be expected in the presence of matrix-fluid attraction or a specific interaction between them. Let us proceed now with the presentation of the results obtained for a pore of spherical symmetry. We have restricted our attention to the case Rc ) 5. For lower radii statistics of the distribution of fluid density would become much worse. We are considering the matrix-filled spherical pore at βµm ) -1.5 and 2, similar to the case of the cylindrical pore for the sake of comparison. The fluid chemical potential, βµf, is 0, 2, and 4. The adsorption isotherms for a fluid adsorbed in the spherical cavity are shown in Figure 4. We have observed that the adsorption is higher for a higher pore microporosity. Moreover, it follows that the adsorption is slightly higher in the spherical pore compared to the cylindrical one under the same values for the chemical potential of matrix species and fluid species in both pores. The density profiles of fluid particles are shown in Figure 5. In Figure 5a the case of dilute matrix (higher microporous spherical pore, P ) 0.943) is considered, whereas in Figure 5b we show the profiles for a high-density matrix (lower microporous spherical pore, P ) 0.716). The oscillations of the density profiles develop in both cases with increasing chemical potential of fluid species. However, the contact values for the profiles, as well as the magnitude of oscillations, are higher in the spherical pore than in the cylindrical one (under similar conditions), as a result of stronger geometric constraints. Moreover, the average adsorbed fluid density in the center of the spherical pore is higher in the spherical pore compared with the cylindrical one. However, similar to the case of adsorption in cylindrical pores, the shape of the density profiles for fluid species is not affected substantially by matrix particle distribution in the absence of attractive or other specific fluid-matrix interactions. The effect of geometric constraints seems to prevail.
Figure 2. Density profiles of particles in cylindrical pore at βµm ) -1.5. Radius of cylinder: Rc ) 1.5 (a), 2.5 (b), 5 (c). The dash-dotted line corresponds to profile of matrix particles, whereas the solid ones correspond to fluid particles at βµf ) 0, 2, or 4, bottom to top.
Structure and Adsorption of a Hard Sphere Fluid
J. Phys. Chem. B, Vol. 102, No. 28, 1998 5493
Figure 4. Adsorption isotherms, Γ*(βµf,βµm), for a fluid in a spherical pore of radius Rc ) 5. The upper and lower curves correspond to βµm ) -1.5, and βµm ) 2, respectively.
Figure 3. Same as in the Figure 2 but at βµm ) 2 and at βµf ) 0, 2, or 3.5, from bottom to top.
Conclusions To conclude, this work considers, we believe for the first time, the problem of adsorption of a simple fluid in matrixfilled pores of cylindrical and spherical geometry. We have restricted ourselves to the model with solely repulsive interparticle interactions and have studied it by using GCMC simulations. We have shown that the adsorption depends on the microporosity values and on geometric constraints due to the pore. Higher adsorption is observed for pores filled by a matrix with higher microporosity and in narrower pores. Layering of the adsorbed fluid in both cylindrical and spherical pores filled with disordered matrix is developed with increasing chemical potential of fluid species. The shape of the density profiles for adsorbed fluid is mostly influenced by constraints due to pore geometry rather than the presence of matrix species in the pore if attractive fluid-matrix interactions are absent in the model. Acknowledgment. This project has been supported in parts by DGAPA of the UNAM under Grant IN111597 and by the
Figure 5. Density profiles of particles in spherical pore of radius Rc ) 5 at βµm ) -1.5 (a) and 2 (b). The dash-dotted line corresponds to the profile of matrix particles, whereas the solid lines correspond to the profiles of fluid particles at βµf ) 0, 2, or 4, bottom to top in both panels.
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